# RZXGate

class qiskit.circuit.library.RZXGate(theta, label=None, *, duration=None, unit='dt')

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Bases: Gate

A parametric 2-qubit $Z \otimes X$ interaction (rotation about ZX).

This gate is maximally entangling at $\theta = \pi/2$.

The cross-resonance gate (CR) for superconducting qubits implements a ZX interaction (however other terms are also present in an experiment).

Can be applied to a QuantumCircuit with the rzx() method.

Circuit Symbol:

     ┌─────────┐
q_0: ┤0        ├
│  Rzx(θ) │
q_1: ┤1        ├
└─────────┘

Matrix Representation:

$\providecommand{\rotationangle}{\frac{\theta}{2}} R_{ZX}(\theta)\ q_0, q_1 = \exp\left(-i \frac{\theta}{2} X{\otimes}Z\right) = \begin{pmatrix} \cos\left(\rotationangle\right) & 0 & -i\sin\left(\rotationangle\right) & 0 \\ 0 & \cos\left(\rotationangle\right) & 0 & i\sin\left(\rotationangle\right) \\ -i\sin\left(\rotationangle\right) & 0 & \cos\left(\rotationangle\right) & 0 \\ 0 & i\sin\left(\rotationangle\right) & 0 & \cos\left(\rotationangle\right) \end{pmatrix}$
Note

In Qiskit’s convention, higher qubit indices are more significant (little endian convention). In the above example we apply the gate on (q_0, q_1) which results in the $X \otimes Z$ tensor order. Instead, if we apply it on (q_1, q_0), the matrix will be $Z \otimes X$:

     ┌─────────┐
q_0: ┤1        ├
│  Rzx(θ) │
q_1: ┤0        ├
└─────────┘
$\providecommand{\rotationangle}{\frac{\theta}{2}} R_{ZX}(\theta)\ q_1, q_0 = exp(-i \frac{\theta}{2} Z{\otimes}X) = \begin{pmatrix} \cos(\rotationangle) & -i\sin(\rotationangle) & 0 & 0 \\ -i\sin(\rotationangle) & \cos(\rotationangle) & 0 & 0 \\ 0 & 0 & \cos(\rotationangle) & i\sin(\rotationangle) \\ 0 & 0 & i\sin(\rotationangle) & \cos(\rotationangle) \end{pmatrix}$

This is a direct sum of RX rotations, so this gate is equivalent to a uniformly controlled (multiplexed) RX gate:

$R_{ZX}(\theta)\ q_1, q_0 = \begin{pmatrix} RX(\theta) & 0 \\ 0 & RX(-\theta) \end{pmatrix}$

Examples:

$R_{ZX}(\theta = 0) = I$ $R_{ZX}(\theta = 2\pi) = -I$ $R_{ZX}(\theta = \pi) = -i Z \otimes X$ $RZX(\theta = \frac{\pi}{2}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & -i & 0 \\ 0 & 1 & 0 & i \\ -i & 0 & 1 & 0 \\ 0 & i & 0 & 1 \end{pmatrix}$

Create new RZX gate.

## Attributes

### base_class

Get the base class of this instruction. This is guaranteed to be in the inheritance tree of self.

The “base class” of an instruction is the lowest class in its inheritance tree that the object should be considered entirely compatible with for _all_ circuit applications. This typically means that the subclass is defined purely to offer some sort of programmer convenience over the base class, and the base class is the “true” class for a behavioural perspective. In particular, you should not override base_class if you are defining a custom version of an instruction that will be implemented differently by hardware, such as an alternative measurement strategy, or a version of a parametrised gate with a particular set of parameters for the purposes of distinguishing it in a Target from the full parametrised gate.

This is often exactly equivalent to type(obj), except in the case of singleton instances of standard-library instructions. These singleton instances are special subclasses of their base class, and this property will return that base. For example:

>>> isinstance(XGate(), XGate)
True
>>> type(XGate()) is XGate
False
>>> XGate().base_class is XGate
True

In general, you should not rely on the precise class of an instruction; within a given circuit, it is expected that Instruction.name should be a more suitable discriminator in most situations.

### condition

The classical condition on the instruction.

### condition_bits

Get Clbits in condition.

### decompositions

Get the decompositions of the instruction from the SessionEquivalenceLibrary.

### definition

Return definition in terms of other basic gates.

### duration

Get the duration.

### label

Return instruction label

### mutable

Is this instance is a mutable unique instance or not.

If this attribute is False the gate instance is a shared singleton and is not mutable.

Return the name.

### num_clbits

Return the number of clbits.

### num_qubits

Return the number of qubits.

### params

The parameters of this Instruction. Ideally these will be gate angles.

### unit

Get the time unit of duration.

## Methods

### inverse

inverse(annotated=False)

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Return inverse RZX gate (i.e. with the negative rotation angle).

Parameters

annotated (bool(opens in a new tab)) –

when set to True, this is typically used to return an

AnnotatedOperation with an inverse modifier set instead of a concrete Gate. However, for this class this argument is ignored as the inverse of this gate is always a RZXGate with an inverted parameter value.

Returns:

RZXGate: inverse gate.

### power

power(exponent, annotated=False)

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Raise this gate to the power of exponent.

Implemented either as a unitary gate (ref. UnitaryGate) or as an annotated operation (ref. AnnotatedOperation). In the case of several standard gates, such as RXGate, when the power of a gate can be expressed in terms of another standard gate that is returned directly.

Parameters

• exponent (float(opens in a new tab)) – the power to raise the gate to
• annotated (bool(opens in a new tab)) – indicates whether the power gate can be implemented as an annotated operation. In the case of several standard gates, such as RXGate, this argument is ignored when the power of a gate can be expressed in terms of another standard gate.

Returns

An operation implementing gate^exponent

Raises

CircuitError – If gate is not unitary